flag

we say that a list of vectors
(u1,…,udn) is an adapted basis relative to the flag, if
the first d1 vectors give a basis of V1, the first d2 vectors
give a basis of V2, etc. Thus, an alternate characterization of a
complete flag, is that the first k elements of an adapted basis are
a basis of Vk.

Example

Let us consider ℝn. For each k=1,…,n let Vk be the
span of e1,…,ek, where ej denotes the jth basic
vector, i.e. the column vector with 1 in the jth position and
zeros everywhere else. The Vk give a complete flag in ℝn .
The list (e1,e2,…,en) is an adapted basis relative to this
flag, but the list (e2,e1,…,en) is not.

Generalizations.

More generally, a flag can be defined as a maximal chain in a partially ordered set. If one considers the poset consisting of subspaces of a (finite dimensional) vector space, one recovers the definition given above.